My B&M Ixon Core headlight can run while charging, but a lot of lights won't do this. I suspect it's hard to design such a thing, or maybe it costs more to make it that way.

I suspect you won't damage anything if you can get your project to work. You may want to test the voltage while it's charging. I don't know what Li-Ion batteries do when the input charging voltage is too low, but I suspect they'll just ignore it, so to speak.